[1]赵中华,王岩青.预处理子空间迭代法[J].东南大学学报(自然科学版),2003,33(4):511-513.[doi:10.3969/j.issn.1001-0505.2003.04.032]
 Zhao Zhonghua,Wang Yanqing.Preconditioning subspace iteration method[J].Journal of Southeast University (Natural Science Edition),2003,33(4):511-513.[doi:10.3969/j.issn.1001-0505.2003.04.032]
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预处理子空间迭代法()
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《东南大学学报(自然科学版)》[ISSN:1001-0505/CN:32-1178/N]

卷:
33
期数:
2003年第4期
页码:
511-513
栏目:
数学、物理学、力学
出版日期:
2003-07-20

文章信息/Info

Title:
Preconditioning subspace iteration method
作者:
赵中华1 王岩青2
1 南京财经大学应用数学系, 南京 210003; 2 解放军理工大学理学院, 南京 210016
Author(s):
Zhao Zhonghua1 Wang Yanqing2
1 Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210003, China
2 Institute of Science, PLA University of Science and Technology, Nanjing 210016, China
关键词:
对称矩阵 特征值 子空间迭代法 预处理技术
Keywords:
symmetric matrix eigenvalues subspace iteration method preconditioning technique
分类号:
O241.6;O175.9
DOI:
10.3969/j.issn.1001-0505.2003.04.032
摘要:
研究了计算大型稀疏对称矩阵的若干个最大或最小特征值的问题.首先引入求解大型对称特征值问题的预处理技术,给出了改善后的算法及相应的算法收敛分析.而求解特征值问题的子空间迭代法,当矩阵的特征值的分布范围较大时,其收敛速度会受到限制.为了加速子空间迭代法的收敛速度,对每次迭代所得的残余矩阵直接进行预处理以改善矩阵特征值的分布而加速收敛.讨论了预处理技术对子空间迭代法的应用,从而给出了预处理子空间迭代法.最后给出了数值例子,结果表明预处理子空间迭代法比子空间迭代法优越,不仅收敛速度快,并且减少了计算量和计算时间.
Abstract:
The problem of computing a few of the largest(or smallest)eigenvalues of a large sparse symmetric matrix is investigated. The preconditioning techniques for computing approximation of the large symmetric matrix are introduced, and improved algorithm and its convergence are presented. The convergence rate of subspace iteration method used to compute eigenvalues problem is confined when the distribution range of eigenvalues is large.In order to accelerate the convergence rate of the subspace iteration method, the preconditioning matrix is used to impact the residual matrix obtained from the iteration procedure, so the distribution of eigenvalues is improved. Having discussed the application of preconditioning techniques to the subspace iteration method, the preconditioning subspace iteration method is presented. Our numerical experiments show that the new method is more effective in convergence of algorithm. And it decreases the computation cost and computation time.

参考文献/References:

[1] Bathe H J,Wilson E L.Solution method for eigenvalue problems in structural mechanics[J].Int J Number Meth Engng, 1973,6:213-226.
[2] 赵中华.用Chebyshev多项式加速的子空间迭代法[J].南京航空航天大学学报,2002,34(2):197-200.
  Zhao Zhonghua.Subspace iteration accelerated by using chebyshev polynomids [J]. J NUAA,2002, 34(2):197-200.(in Chinese)
[3] Davidson E R.The iterative calculation of a few of the lowest eigenvalue and corresponding eigenvectors of large real symmetric matrices[J].J Comput Phys,1975,17:87-94.
[4] Crouzeix M,Philippe B,Sadkane M.The Davidson method [J].SIAM J Sci Computing,1994,15(1):62-76.
[5] Morgan R B,Scott D S.Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems[J].SIAM J Sci Computing,1993,14:585-593.
[6] Dai Hua,Lancaster P.Preconditioning block Lanczos algorithm for solving symmetric eigenvalue problems[J].J Comput Math,2000,18(4):365-374.

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备注/Memo

备注/Memo:
作者简介: 赵中华(1976—), 女, 硕士, 助教, zhonghua-zhao@sina.com.
更新日期/Last Update: 2003-07-20